Vilfredo Pareto and Multi-objective Optimization - Department of ...

Documenta Math. 447 

Vilfredo Pareto and Multi-objective Optimization 

Matthias Ehrgott 

2010 Mathematics Subject Classification: 90C29 

KeywordsandPhrases: Multi-objectiveoptimization, Paretooptimality 

A multi-objective optimization problem consists in the simultaneous optimization 

of p objective functions f 1 ,...,f p subject to some constraints, which I will 

just write as x ∈ X, where X is a subset of R n . It is usually assumed that there 

does not exist any x ∈ X such that all functions f k attain their minimima at x. 

Hence, due to the absence of a total order on R p , it is necessary to define the 

minimization with respect to partial orders. So let Y := {f(x) : x ∈ X} be the 

set of outcome vectors. To compare elements of Y, I will follow the definition 

of Koopmans (1951). Let y 1 ,y 2 ∈ Y. Then y 1 ≦ y 2 if and only if yk 1 ≦ y2 k for 

all k = 1,...p; y 1 ≤ y 2 if and only if y 1 ≦ y 2 , but y 1 ≠ y 2 and y 1 < y 2 if and 

only if yk 1 < y2 k 

for all k = 1,...p. 

It is here that Pareto makes his appearance. In countless books and articles 

on multi-objective optimization, one can find a definition like this: 

Definition 1. Let X ⊂ R n be a non-empty set of feasible solutions and f = 

(f 1 ,...f p ) : R n → R p be a function. Feasible solution ˆx ∈ X is called a Pareto 

optimal solution of the multi-objective optimization problem 

min{f(x) : x ∈ X} (1) 

if and only if there does not exist any x ∈ X such that f(x) ≤ f(ˆx). 

Sometimes Pareto optimality is defined with respect to outcome vectors. 

Definition 2. Let Y ∈ R p be a non-empty set of outcome vectors. Outcome 

vector ŷ ∈ Y is called Pareto optimal if and only if there does not exist any 

y ∈ Y such that y ≤ ŷ. 

Where does the name Pareto optimal come from? Vilfredo Pareto and Francis 

Ysidro Edgeworth are often called as the fathers of multi-objective optimization. 

Sentences like the “introduction of the Pareto optimal solution in 

1896” (Chen et al., 2005, p. VII); “The concept of noninferior solution was introduced 

at the turn of the century [1896] by Pareto, a prominent economist” 

(Chankong and Haimes, 1983, p. 113); “Edgeworth and Pareto were probably 

Documenta Mathematica · Extra Volume ISMP (2012) 447–453

448 Matthias Ehrgott 

the first who introduced an optimality concept for such problems” (Jahn, 2004, 

p. 113); “wurden besonders von F.Y. Edgeworth (1845–1926) and V. Pareto 

(1848–1929 [sic!]) hinreichende Bedingungen für Paretomaximalität bzw. Gleichgewichtsbedingungen 

angegeben.” (Göpfert and Nehse, 1990, p. 9) or “The 

foundations are connected with the names of Vilfredo Pareto (1848–1923) and 

Francis Ysidro Edgeworth (1845–1926)” (Löhne, 2011, p. 1) abound in textbooks. 

TheInternationalSocietyonMultipleCriteriaDecisionMakingbestows 

the Edgeworth–Pareto award “upon a researcher who, over his/her career, has 

established a record of creativity to the extent that the field of MCDM would 

notexistinitscurrentformwithoutthefar-reachingcontributionsfromthisdistinguished 

scholar”, see http://www.mcdmsociety.org/intro.html#Awards. 

Edgeworth was an influential Professor of Economics at King’s College London 

and from 1891 Professor of Political Economy at Oxford University. In his 

best known book Mathematical Psychics (Edgeworth, 1881) he applied formal 

mathematics to decision making in economics. He developed utility theory, 

introducing the concept of indifference curve and is best known for the Edgeworth 

box. But because multi-objective optimization is concerned with Pareto 

optimality rather than Edgeworth optimality, this story focuses on his contemporary. 

Fritz Wilfried Pareto 

According to Yu (1985, p. 49) Pareto “was a famous Italian engineer” but he 

is certainly much better known as an economist. The following information 

is taken from Stadler (1979) and the wikipedia entry (http://en.wikipedia. 

org/wiki/Vilfredo_Pareto) on Pareto. 

Vilfredo Federico Damaso Pareto was born on 15 July 1848 in Paris as Fritz 

Wilfried Pareto, son of a French woman and an Italian civil engineer, who was 

a supporter of the German revolution of 1848. His name was changed to the 

Italian version when his family moved back to Italy in 1855 (or 1858). In 1870 

he graduated from Polytechnic Institute of Turin with a dissertation entitled 

“The Fundamental Principles of Equilibrium in Solid Bodies”. He then worked 

as an engineer and manager for an Italian railway company. He was very 

politically active, an ardent supporter of free market economy. He obtained a 

lecturer position in economics and management at the University of Florence 

in 1886 (according to wikipedia). Eventually he resigned from his positions in 

1889. During the 1880s he became acquainted with leading economists of the 

time and he published many articles by 1893 (not all academic, though). In 

1893 he moved to Lausanne where he lectured at the University of Lausanne 

and became the successor of Léon Walras as Professor of Political Economy. In 

his later years he mainly worked in Sociology. Vilfredo Pareto died at Célégny, 

Switzerland, on 19 August 1923. The University of Lausanne still has a Centre 

d’études interdisciplinaires Walras Pareto (http://www.unil.ch/cwp). Apart 

from Pareto optimality, Pareto’s name is attached to the Pareto principle (or 

80–20 rule), observing in 1906 that 80% of the property in Italy was owned by 


Vilfredo Pareto and Multi-objective Optimization 449 

Figure 1: Vilfredo Pareto 1848–1923 (Picture scanned from the second French 

edition of Pareto (1906) published in 1927.) 

20% of the population and the Pareto distribution, a power law probability 

distribution. 

Pareto Optimality 

The origin of the term Pareto optimality goes back to the following text from 

Pareto (1906, Chapter VI, Section 33). 

Principeremo col definire un termine di cui è comodo fare uso per 

scansare lungaggini. Diremo che i componenti di una collettività 

godono, in una certa posizione, del massimo di ofelimità, quando 

è impossibile allontanarsi pochissimo da quella posizione giovando, 

o nuocendo, a tutti i componenti la collettività; ogni piccotissimo 

spostamento da quella posizione avendo necessariamente per effetto 

digiovareapartedeicomponentitacollettivitàedinuocereadaltri. 

Or in the English translation (Pareto, 1971, p. 261): 



Wewillbeginbydefiningatermwhichisdesirabletouseinorderto 

avoid prolixity. We will say that the members of a collectivity enjoy 

maximum ophelimity in a certain position when it is impossible to 

find a way of moving from that position very slightly in such a 

manner that the ophelimity enjoyed by each of the individuals of 

that collectivity increases or decreases. That is to say, any small 

displacement in departing from that position necessarily has the 

effect of increasing the ophelimity which certain individuals enjoy, 

and decreasing that which others enjoy, of being agreeable to some 

and disagreeable to others. 

Of course, Pareto here refers to the distribution of utility (ophelimity) among 

individuals in an economy rather than solutions of an optimization problem. 

Multi-objective optimization or mathematical optimization in general as we 

know it today, did not exist during Pareto’s lifetime, it only developed in 

the 1940s. And it is some of the founding works of Operations Research and 

optimization that need to be cited here. Nobel Laureate in Economics T.C. 

Koopmans (1951) formally studied production as a resource allocation problem 

and the combination of activities to represent the output of commodities as a 

function of various factors. In this work he introduced the following definition 

of efficient vector (p. 60). “A point y in the commodity space is called efficient 

if it is possible [i.e., if y ∈ (A)], and if there exists no possible point ȳ ∈ (A) such 

that ȳ −y ≥ 0.” Note that (A) is what I called Y in Definition 2, i.e., possible 

means that there is some x such that y = Ax. Koopmans does hence only talk 

about efficient vectors in terms of the outcome set. He proves necessary and 

sufficient conditions for efficiency, but he does not refer to Pareto, nor does he 

talk about Pareto optimal solutions as in Definition 1 – instead he refers to “an 

activity vector x (that) shall lead to an efficient point y = Ax”. 

Another classic reference in optimization is the seminal paper by 

Kuhn and Tucker (1951). They refer to the “vector maximum of Koopmans’ 

efficient point type for several concave functions g 1 (x),...,g p (x)”. This 

seems to be the earliest reference to the optimization of several functions in 

mathematics. Kuhn and Tucker cite Koopmans (1951) when they talk about 

vector maximum. They also apply the term efficient to the solutions of vector 

optimization problems (i.e., in decision space) and introduce the notion of 

proper efficiency. But, again, no mention of Pareto. Kuhn and Tucker (1951) 

cite another Nobel Laureate in Economics who contributed to the foundations 

of multi-objective optimization, Kenneth J. Arrow. 

Arrow discusses Pareto extensively in his economical work and statements of 

the impossibility theorem today usually refer to Pareto optimality as one of the 

axioms that cannot be jointly satisfied by a social choice function, but this term 

does not appear in Arrow’s original formulation (Arrow, 1951). Arrow’s important 

contribution to multi-objective optimization (Arrow et al., 1953) starts as 

follows “A point s of a closed convex subset S of k-space is admissible if there is 

no t ∈ S with t i ≤ s i for all i = 1,...,k, t ≠ s.” This is, of course, the same as 



Koopmans’ definition of efficient point (whose paper Arrow et al. (1953) cite), 

and again is relevant in the outcome set of a multi-objective problem rather 

than the set of feasible solutions – no trace of Pareto here, either. 

There are a number of other definitions of Pareto optimal, efficient, or admissible 

points. Zadeh (1963) defines “A system S 0 ∈ C is noninferior in C if the 

intersection of C and Σ > (S 0 ) is empty.” Σ > (S 0 ) is the set of all systems which 

are better than S 0 with respect to a partial order ≥. Chankong and Haimes 

(1983) later use the same definition. While Zadeh cites Koopmans and Kuhn 

and Tucker, Pareto remains notably absent. The final term that is common 

today is that of a nondominated point. 

Multiobjective Optimization and Economics 

When did the term Pareto optimal first appear in the literature? As we have 

seen, it was not used in early mathematical works on multi-objective optimization. 

The answer is once again in economics. Little (1950, p. 87) in a discussion 

of the distribution of income (i.e., in the same context as Pareto himself) uses 

the term Pareto ‘optimum’ (with the quotation marks). The origin of the term 

is, therefore, clearly found in economics. It has then apparently mostly been 

used in economics, appearing in journals such as Public Choice and Journal 

of Economic Theory. As shown above, it was not used by the economists 

who are credited with having contributed to the origins of the mathematical 

theory of multi-objective optimization, but migrated to mathematics later on. 

The first journal articles that I could find are Basile and Vincent (1970) and 

Vincent and Leitmann (1970). These articles also used the term undominated 

as an alternative. This then turned to nondominated in Yu and Leitmann 

(1974). 

Economics had a strong influence on the early history of multi-objective optimization, 

especially Pareto’s original definition of the term maximum ophelimity 

and the origin of the term Pareto optimum in Little (1950). The move 

into mathematics and optimization coincides with the mathematization of economics 

by scholars such as Koopmans and Arrow and finally the introduction 

of the topic into mathematical optimization by Kuhn and Tucker. It seems 

to have taken quite a while for Pareto’s name to appear in the mathematical 

optimization literature. 

The consequence of the history of Pareto optimality outlined above, is that 

at present there are quite a few terms (efficient, noninferior, nondominated, 

admissible, Pareto optimal) that express the same idea. Since multi-objective 

optimization often distinguishes between decision vectors x ∈ X and outcome 

vectors y ∈ Y, one can find a large number of combinations of these terms 

in the literature used in parallel today, such as Pareto optimal decisions and 

efficient outcomes. 

Itturnsoutthatthehistoryofmulti-objectiveoptimization(vectoroptimization) 

is quite an interesting read, and I would like to refer interested readers 

to Stadler (1979) as a starting point. The history of multiple criteria deci- 



sion making in general is the topic of the book Köksalan et al. (2011). These 

works also consider roots of multi-objective optimization in game theory and 

the theory of ordered spaces and vector norms. 

References 

K. J. Arrow. Social Choice and Individual Values. Cowles Commission for 

Research in Economics Monograph No. 12. John Wiley & Sons, New York, 

1951. 

K. J. Arrow, E. W. Barankin, and D. Blackwell. Admissible points of convex 

sets. In H.W. Kuhn and A.W. Tucker, editors, Contributions to the Theory 

of Games, volume 2, pages 87–91. Princeton University Press, Princeton, 

1953. 

G. Basile and T. L. Vincent. Absolutely cooperative solution for a linear, multiplayer 

differential game. Journal of Optimization Theory and Applications, 

6:41–46, 1970. 

V. Chankong and Y. Y. Haimes. Multiobjective Decision Making – Theory and 

Methodology. Elsevier Science, New York, 1983. 

G. Chen, X. Huang, and X. Yang. Vector Optimization – Set-Valued and Variational 

Analysis, volume 541 of Lecture Notes in Economics and Mathematical 

Systems. Springer Verlag, Berlin, 2005. 

F. Y. Edgeworth. Mathematical Psychics. C. Kegan Paul & Co., London, 1881. 

A. Göpfert and R. Nehse. Vektoroptimierung, volume 74 of Mathematisch- 

Naturwissenschaftliche Bibliothek. BSB B.G. Teubner Verlagsgesellschaft, 

Leipzig, 1990. 

J. Jahn. Vector Optimization – Theory, Applications, and Extensions. Springer 

Verlag, Berlin, 2004. 

M. Köksalan, J. Wallenius, and S. Zionts. Multiple Criteria Decision Making 

– From Early History to the 21st Century. World Scientific Publishing, 

Singapore, 2011. 

T. C. Koopmans. Analysis of production as an efficient combination of activities. 

In T.C. Koopmans, editor, Activity Analysis of Production and Allocation, 

Cowles Commission for Research in Economics Monograph No. 13, 

pages 33–97. John Wiley & Sons, New York, 1951. 

H. W. Kuhn and A. W. Tucker. Nonlinear programming. In J. Neyman, editor, 

Proceedings of the Second Berkeley Symposium on Mathematical Statistics 

and Probability, pages 481–492. University of California Press, Berkeley, 

1951. 



I.M.D.Little. A Critique of Welfare Economics. TheClarendonPress, Oxford, 

1950. 

A. Löhne. Vector Optimization with Infimum and Supremum. Springer Verlag, 

Berlin, 2011. 

V. Pareto. Manuale di Economia Politica. Società Editrice Libraria, Milan, 

1906. 

V. Pareto. Manual of Political Economy. Augustus M. Kelley Publishers, New 

York, 1971. 

W. Stadler. A survey of multicriteria optimization or the vector maximum 

problem, Part I: 1776-1960. Journal of Optimization Theory and Applications, 

29:1–52, 1979. 

T. L. Vincent and G. Leitmann. Control-space properties of cooperative games. 

Journal of Optimization Theory and Applications, 6:91–113, 1970. 

P. L. Yu. Multiple Criteria Decision Making: Concepts, Techniques and Extensions. 

Plenum Press, New York, 1985. 

P. L. Yu and G. Leitmann. Compromise solutions, domination structures, and 

Salukvadze’s solution. Journal of Optimization Theory and Applications, 13: 

362–378, 1974. 

L. A. Zadeh. Optimality and non-scalar-valued performance criteria. IEEE 

Transactions on Automatic Control, 8:59–60, 1963. 

Matthias Ehrgott 

DepartmentofEngineeringScience 

The University of Auckland 

New Zealand 

m.ehrgott@auckland.ac.nz 


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